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Class 11 RD Sharma Solutions – Chapter 31 Derivatives – Exercise 31.3
  • Last Updated : 21 Dec, 2020

Question 1. Find the component statements of the following compound statements:

(i) The sky is blue, and the grass is green.

(ii) The earth is round, or the sun is cold.

(iii) All rational numbers are real, and all real numbers are complex.

(iv) 25 is a multiple of 5 and 8.

Solution:



Every compound statement has components .

(i) The component’s of the statement are:

A: The sky is blue.

B: The grass is green.

(ii) The component’s of the statement are:

A: The earth is round.

B: The sun is cold.

(iii) The component’s statement are:

A: All rational number are real.

B: All real number are complex.

(iv) The component’s of the statement are:

A: 25 is multiple of 5.

B: 25 is multiple of 8.

Question 2. For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer.

(i) Students can take Hindi or Sanskrit as their third language.

(ii) To entry a country, you need a passport or a voter registration card.

(iii) A lady gives birth to a baby boy or a baby girl.

(iv) To apply for a driving license, you should have a ration card or a passport.

Solution:



(i) A choice is given to a student to choose either Hindi or Sanskrit as their third language. The student can’t choose both. Therefore, an exclusive “OR” is used because a student cannot take both Hindi and Sanskrit.

(ii) A passport or voter registration card is needed by a person to enter a country. However, a person may have both. Therefore, an inclusive “OR” is used because a person can have both a passport and a voter registration card to enter a country.

(iii) A lady cannot give birth to a baby who is both a boy and a girl, therefore, an exclusive “OR” is used because either a boy or girl is born.

(iv) A ration card  or passport is needed to apply a driving license. However, a person may have both.Therefore, an inclusive “OR” is used because a person can have both a ration card and passport to apply for a driving license.

Question 3. Write the component statements of the following compound statements and check whether the compound statement is true or false:

(i) To enter into a public library children need an identification card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) x = 2 and x = 3 are the roots of the equation 3x2 – x – 10 = 0.

(v) The sand heats up quickly in the sun and does not cool down fast at night.

Solution:

(i) The components are:

P: To get into a public library children need an identity card from the school.

Q: To get into a public library children need a letter from the school authorities.

Both of the statements P and Q are true. Therefore, the compound statement is true.

(ii) The components are:

P: All rational numbers are real.

Q: All real numbers are not complex.

The statement P is true and Q is false therefore, P and Q both together are False. Therefore, the compound statement is False

(iii) The components are:

P: Square of an integer is positive.

Q: Square of an integer is negative.

Both statements P and Q are true. Therefore, the compound statement is True.

(iv) The components of the compound statement are:

P: x = 2 is a root of the equation 3x2 – x – 10 = 0

Q: x = 3 is a root of the equation 3x2 – x – 10 = 0

The statement P is true, but Q is false then P and Q both combined are False. Therefore, the compound statement is False.

(v) The components of the compound statement are:

P: The sand heats up quickly in the sun.

Q: The sand does not cool down fast at night.

Both statements P and Q both are False. Therefore, the compound statement is False.

Question 4. Determine whether the following compound statements are true or false:

(i) Delhi is in India and 2 + 2 = 4

(ii) Delhi is in England and 2 + 2 = 4

(iii) Delhi is in India and 2 + 2 = 5

(iv) Delhi is in England and 2 + 2 = 5

Solution:

(i) The components are:

A: Delhi is in India.

B: 2 + 2 = 4

Both statements A and B are true. Hence, the compound statement is True.

(ii) The components are:

A: Delhi is in England.

B: 2 + 2 = 4

The statement A is false, and B is true. So, both A and B together are false. Hence, the compound statement is False.

(iii) The components are:

A: Delhi is in India.

B: 2 + 2 = 5

The statement B is false, and A is true. So, both A and B together are false. Hence, the compound statement is False.

(iv) The components are:

A: Delhi is in England.

B: 2 + 2 = 5

Both the statements A and B are false. Hence, the compound statement is False.

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